let f1, f2 be Function; :: thesis: ( dom f1 = PARTITIONS Y & ( for x being set st x in PARTITIONS Y holds
ex PA being a_partition of Y st
( PA = x & f1 . x = ERl PA ) ) & dom f2 = PARTITIONS Y & ( for x being set st x in PARTITIONS Y holds
ex PA being a_partition of Y st
( PA = x & f2 . x = ERl PA ) ) implies f1 = f2 )
assume that
A3: dom f1 = PARTITIONS Y and
A4: for x being set st x in PARTITIONS Y holds
ex PA being a_partition of Y st
( PA = x & f1 . x = ERl PA ) and
A5: dom f2 = PARTITIONS Y and
A6: for x being set st x in PARTITIONS Y holds
ex PA being a_partition of Y st
( PA = x & f2 . x = ERl PA ) ; :: thesis: f1 = f2
A7: for z being set st z in PARTITIONS Y holds
f1 . z = f2 . z

proof

let x be set ; :: thesis: ( x in PARTITIONS Y implies f1 . x = f2 . x )
assume A8: x in PARTITIONS Y ; :: thesis: f1 . x = f2 . x
A9: ( ex PA being a_partition of Y st
( PA = x & f1 . x = ERl PA ) & ex PA being a_partition of Y st
( PA = x & f2 . x = ERl PA ) ) by A4, A6, A8;
thus f1 . x = f2 . x by A9; :: thesis: verum

end;

thus f1 = f2 by A3, A5, A7, FUNCT_1:9; :: thesis: verum